
\(\bullet \)  \(f_{1}(x)=x^{p^{k}+1}\) , where \(k \ge 0\)  and \( 2\nmid \frac{m}{{\rm gcd}(m,k)}\)  (Dembowski and Ostrom [1]}).

\(\bullet \)  \(f_{2}(x)=x^{\frac{p^{k}+1}{2}}\) , where \(p=3\) , \( 2\nmid k\) , and \({\rm gcd}(m,k)=1\)  (Coulter and Matthews [2]}).

\(\bullet \)  \(f_{3}(x)=x^{10}-\beta x^{6}-\beta ^{2}x^{2}\) , where \(p=3\) , \( 2\nmid m\)  and \(\beta \in \mathbb {F}_{p^m}^{*}\)  (Ding and Yuan [3]}).

\(\bullet \)  \(f_{4}(x)=\beta x^{p^k+1}-\beta ^{p^s}x^{p^{ls}+p^{-ls+k}}\) , where \(\beta \)  is a primitive element in \(\mathbb {F}_{p^m}\) , \(m=3s\) , \({\rm gcd}(3,s)=1\) , \(2\nmid \frac{s}{{\rm gcd}(s,k)}\) , \(k\equiv \pm s \pmod {3}\) , \(l=1\)  if \(s-k=0 \pmod {3}\)  and \(l=-1\)  if \(s+k=0 \pmod {3}\)  (Zha, Kyureghyan and Wang [4]}).

\(\bullet \)  \(f_{5}(x)=(\beta x)^{p^k+1}-((\beta x)^{p^k+1})^{p^s}+\sum _{i=0}^{s-1}c_{i}x^{p^i(p^s+1)}\) , where \(m=2s\) , both \(s\)  and \(k\)  are positive integers, \({\rm gcd}(s+k,2s)={\rm gcd}(s+k,s)\) , \({\rm gcd}(p^k+1,p^s+1)\ne {\rm gcd}(p^k+1,\frac{p^s+1}{2}),\) 
\(\beta \in \mathbb {F}_{p^m}^{*}\)  and \(\sum _{i=0}^{s-1}c_{i}x^{p^i}\)  is a permutation polynomial of \(\mathbb {F}_{p^m}\)  (Budaghyan and Helleseth [5]}).

\(\bullet \)  \(f_{6}(x)=\beta x^{p^s+1}+zx^{p^k+p^t}+z^{p^s}x^{p^{k+s}+p^{s+t}}+\sum _{i=0}^{s-1}w_{i}x^{p^i(p^s+1)}\) , where \(m=2s\) , both \(s\)  and \(k\)  are positive integers, \(w_{i} \in \mathbb {F}_{p^s}\) , \( 2\nmid \frac{m}{{\rm gcd}(m,t-k)}\) , \(\beta \in \mathbb {F}_{p^m}\setminus \mathbb {F}_{p^s}\)  and \(z=\alpha ^{r}\)  for \(\alpha \)  being a primitive element in \(\mathbb {F}_{p^m}\)  and \({\rm gcd}(p^{k-t}+1,p^s+1)\nmid r\)  (Budaghyan and Helleseth [5]}).

\(\bullet \)  \(f_{7}(x)=x^{p^t+1}-\beta x^{p^{2s}+p^{s+t}}\) , where \(m=3s\) , \(s^\prime =\frac{s}{{\rm gcd}(s,t)}\)  is odd, \(t^\prime =\frac{t}{{\rm gcd}(s,t)}\) , \({\rm ord}(\beta )=p^{2s}+p^s+1\) , \(t^\prime +s^\prime \equiv 0\pmod {3}\)  or \(p^s\equiv p^t\pmod {3}\)  (Bierbrauer [7]}).

\(\bullet \)  \(f_{8}(x)=x^{p^t+1}-\beta x^{p^{3s}+p^{t+s}}\) , where \(m=4s\) , \(2\nmid \frac{2s}{\gcd (2s,t)}\) , \(p^s\equiv p^t\equiv 1\pmod {4}\) , \({\rm ord}(\beta )=p^{3s}+p^{2s}+p^s+1\)  (Bierbrauer [7]}).

